Points (2 ,1 ) and (5 ,9 ) are (3 pi)/4 radians apart on a circle. What is the shortest arc length between the points?

1 Answer
Oct 12, 2016

s ~~ 10.89

Explanation:

The length of the line segment, c, between the two points points is:

c = sqrt((5 - 2)^2+(9 - 1)^2)

c = sqrt(3^2+8^2)

c = sqrt(73)

Because the line segment between the two points and two radii form a triangle, we can use the Law of Cosines to find the radius:

c^2 = r^2 + r^2 - 2(r)(r)cos(theta)

r = sqrt(c^2/(2 - 2cos(theta))

The arc length, s, is found using the following:

s = rtheta

s = thetasqrt(c^2/(2 - 2cos(theta))

s = (3pi/4)sqrt(73/(2 - 2cos(3pi/4))

s ~~ 10.89